ene gies
A icle
Powe Beacon-Assis ed Ene gy Ha es ing in a
Hal -Duplex Communica ion Ne wo k unde
Co-Channel In e e ence o e a Rayleigh Fading
En i onmen : Ene gy E iciency and Ou age
P obabili y Analysis
Van-Duc Phan 1, Tan N. Nguyen 2,* , Minh T an 3, T an Thanh T ang 4, Mi osla Voznak 5,
Duy-Hung Ha 2,5 and Thanh-Long Nguyen 6
1
Cen e o Excellence o Au oma ion and P ecision Mechanical Enginee ing, Nguyen Ta Thanh Uni e si y,
Ho Chi Minh Ci y 70000, Vie nam
2
Wi eless Communica ions Resea ch G oup, Facul y o Elec ical & Elec onics Enginee ing, Ton Duc Thang
Uni e si y, 19 Nguyen Huu Tho S ee , Ho Chi Minh Ci y 70000, Vie nam
3Op oelec onics Resea ch G oup, Facul y o Elec ical and Elec onics Enginee ing, Ton Duc Thang
Uni e si y, Ho Chi Minh Ci y 70000, Vie nam
4Na ional Key Labo a o y o Digi al Con ol and Sys em Enginee ing, Ho Chi Minh Ci y 70000, Vie nam
5Facul y o Elec ical Enginee ing and Compu e Science, Technical Uni e si y o Os a a, Os a a 70833,
Czech Republic
6Cen e o In o ma ion Technology, Ho Chi Minh Ci y Uni e si y o Food Indus y,
Ho Chi Minh Ci y 70000, Vie nam
*Co espondence: [email p o ec ed]
Recei ed: 30 May 2019; Accep ed: 1 July 2019; Published: 4 July 2019


Abs ac :
In his ime, ene gy e iciency (EE), measu ed in bi s pe Wa , has been conside ed as
an impo an eme ging me ic in ene gy-cons ained wi eless communica ion ne wo ks because o
hei ene gy sho age. In his pape , we in es iga e powe beacon assis ed (PB) ene gy ha es ing
(EH) in hal -duplex (HD) communica ion ne wo k unde co-channel In e e e o e Rayleigh ading
en i onmen . In his wo k, we in es iga e he model sys em wi h he ime swi ching (TS) p o ocol.
Fi s ly, he exac and asymp o ic o m exp essions o he ou age p obabili y (OP) a e analyzed and
de i ed. Then he sys em EE is in es iga ed and he in luence o he p ima y sys em pa ame e s on
he sys em pe o mance. Finally, we e i y he co ec ness o he analy ical exp essions using Mon e
Ca lo simula ion. Finally, we can s a e ha he simula ion and analy ical esul s a e he same.
Keywo ds:
powe beacon (PB); ene gy e iciency (EE); ou age p obabili y (OP); ene gy ha es ing
(EH); co-channel in e e ence (CCI)
1. In oduc ion
Nowadays, he In e ne o Things (IoT) is conside ed o be a ho esea ch opic. In his ne wo k,
all sma de ices can wo k and coope a e o e he In e ne en i onmen . The links be ween he digi al
and eal wo lds help sma de ices o eac like humans (sma de ices can hea , see, hink, make
decisions, and pe o m complica ed asks). The c i ical ad an age o he IoT is ha sma de ices
can au oma ically do hei asks wi hou human in ol emen . Nowadays, he IoT is in ol ed in all
aspec s o ci ilian li e, including anspo a ion, sma g ids, secu i y and public sa e y, ag icul u e,
logis ics, and e-heal h. Fu he mo e, ene gy ha es ing (EH), in which adio equency (RF) signals can
ans e bo h in o ma ion and ene gy a he same ime, can be conside ed o be a p omising solu ion
Ene gies 2019,12, 2579; doi:10.3390/en12132579 www.mdpi.com/jou nal/ene gies
Ene gies 2019,12, 2579 2 o 14
o ex ending he li e ime o ene gy-cons ained IoT de ices [
1
–
10
]. EH communica ion sys ems
commonly use he wo adi ional ime swi ching (TS) and powe spli ing (PS) p o ocols, wi h he
compa ison o hese p o ocols in di e en sys em ne wo ks. Mo eo e , we can see ha he loss o
in o ma ion in he ha es ing phase in he TS p o ocol and he low co e age a ea in he PS p o ocol a e
he main disad an ages o hese p o ocols. In addi ion, he complica ed ha dwa e s uc u e o he PS
p o ocol compa ed wi h he simple ha dwa e in he TS p o ocol is also a main disad an age o he PS
p o ocol. [
11
–
16
]. In his esea ch, we selec ed he hal -duplex (HD) model because o i s simplici y
and he possibili y o applying i in eal-wo ld scena ios. In eali y, no all wi eless nodes a e equipped
wi h ull-duplex capaci y due o limi a ions in ha dwa e o implemen a ion cos . F om his poin o
iew, we conside ed he TS p o ocol and he HD model o ou model sys em.
In he nex gene a ion o communica ion sys ems, billions o wi eless de ices will be connec ed
o each o he ia he IoT sys em, which will lead o an ene gy consump ion p oblem. The e o e,
ene gy e iciency (EE), measu ed in bi s pe wa , is conside ed o be an impo an eme ging me ic in
ene gy-cons ained wi eless communica ion ne wo ks due o hei ene gy sho age [
17
–
19
]. The au ho s
o [
20
] in es iga ed he EE op imiza ion p oblems in he SWIPT senso ne wo ks, which employs bo h
he PS and he TS p o ocols. In [
21
], he au ho s in es iga ed an EE esou ce alloca ion algo i hm
o SWIPT in an OFDMA sys em, whe e he ecei e s employed he PS scheme o ha es ene gy.
The au ho s o [
22
] in es iga ed OFDMA sys ems wi h he PS SWIPT, and p oposed a esou ce
alloca ion (ResAll) scheme o maximize EE. Wi hou conside ing SWIPT, o he MIMO wo-way
AF elay channels, he la es EE beam o ming scheme was de i ed in [
23
], which only conside s a
single-s eam ansmission pe use and canno be applied o mul iple da a s eam scena ios because
o he di e en objec i e exp essions. In [
24
], he op imal global EE pe o mance was in es iga ed in
in e e ence-limi ed ne wo ks whe e he SWIPT scheme was no employed. The EE in SWIPT is based
on he IoT dis ibu ed an enna sys em (DAS), as s udied in [
25
]. In [
26
], a ime-slo ed la ge-scale MIMO
sys em o ene gy ha es ing and in o ma ion ansmission was s udied, whe e an ene gy-e icien
op imiza ion scheme was p oposed by join ly op imizing bo h ans e du a ion and ansmi ed powe .
The au ho s o [
27
] de eloped a dis ibu ed i e a ion algo i hm o powe alloca ion, PS a io, and
elay selec ion o maximize EE in clus e ed wi eless senso ne wo ks wi h SWIPT. The au ho s o [
28
]
conside ed he EE maximiza ion p oblem in a wi eless powe ed communica ion ne wo k (WPCN),
whe e mul iple use s ha es ene gy om a dedica ed powe s a ion and hen communica e wi h an
in o ma ion ecei ing s a ion.
In his esea ch, we p opose and in es iga e powe beacon-assis ed (PB) EH in an HD
communica ion ne wo k unde co-channel in e e ence o e a Rayleigh ading en i onmen . We
conside ed he model sys em o be he TS p o ocol. Fi s ly, he exac and asymp o ic o m exp essions
o he ou age p obabili y (OP) we e de i ed. Then we in es iga ed he EE o he model sys em and he
in luence o he p ima y sys em pa ame e s on he pe o mance o he p oposed sys em. Finally, he
accu acy o he analy ical exp essions was de i ed and e i ied using a Mon e Ca lo simula ion in
connec ion wi h he p ima y sys em pa ame e s. F om he esul s, we de e mined ha he analy ical
ma hema ical and simula ed esul s a e he same. The main con ibu ions a e summa ized as ollows:
1.
The sys em model o a PB EH in an HD communica ion ne wo k unde co-channel in e e ence
o e a Rayleigh ading en i onmen .
2. The exac and asymp o ic o m exp essions o he OP we e de i ed.
3.
The EE o he model sys em and he in luence o he p ima y sys em pa ame e s on he pe o mance
o he p oposed sys em we e in es iga ed.
4.
A Mon e Ca lo simula ion was conduc ed o e i y he analysis esul s using he p ima y
sys em pa ame e s.
The es o his pape is o ganized as ollows. Sec ion 2p esen s he sys em model. Sec ion 3
in es iga es he sys em pe o mance and he sys em EE. Sec ion 4gi es he esul s, and some discussions
a e p o ided. Finally, some conclusions a e d awn in Sec ion 5.
Ene gies 2019,12, 2579 3 o 14
2. Sys em Model
In his wo k, we conside a PB EH in an HD communica ion ne wo k. F om Figu e 1, we
deno e ha S and D a e he sou ce node and he des ina ion node, espec i ely, PB deno es a powe
beacon-assis ed node, and I deno es he co-channel in e e ence om he en i onmen . Le
1
and
2
ep esen he I-S and I-D in e e ence channels, espec i ely. Le hand gdeno e he PB-S and S-D
channels. Assume ha all channels a e Rayleigh block ading channels. As d awn in Figu e 2, he
whole ansmission block (T) can be di ided in o pa s. Le T and 0 <
α
<1 deno e he whole symbol
du a ion and he TS ac o , espec i ely. S and I sca enge ene gy om he adio equency signal
ecei ed om he PB node S and node I, espec i ely, du ing αT. Mo eo e , he emaining (1 −α)T is
spen on signal ansmission om node S o node D, and he signal om node I o node D. No e ha all
he ene gy ha es ed a node S and node I is consumed o o wa ding sou ce in o ma ion o node D.
Ene gies 2019, 12, x FOR PEER REVIEW 3 o 15
2. Sys em Model
In his wo k, we conside a PB EH in an HD communica ion ne wo k. F om Figu e 1, we deno e
ha S and D a e he sou ce node and he des ina ion node, espec i ely, PB deno es a powe beacon-
assis ed node, and I deno es he co-channel in e e ence om he en i onmen . Le 1 and 2 ep esen
he I-S and I-D in e e ence channels, espec i ely. Le h and g deno e he PB-S and S-D channels.
Assume ha all channels a e Rayleigh block ading channels. As d awn in Figu e 2, he whole
ansmission block (T) can be di ided in o pa s. Le T and 0 < α < 1 deno e he whole symbol du a ion
and he TS ac o , espec i ely. S and I sca enge ene gy om he adio equency signal ecei ed
om he PB node S and node I, espec i ely, du ing αT. Mo eo e , he emaining (1 − α)T is spen on
signal ansmission om node S o node D, and he signal om node I o node D. No e ha all he
ene gy ha es ed a node S and node I is consumed o o wa ding sou ce in o ma ion o node D.
IT
EH
PB S D
I
Figu e 1. Sys em model. PB, powe beacon-assis ed node; EH, ene gy ha es ing; I, co-channel
in e e ence om he en i onmen ; IT, in o ma ion ansmission; S, sou ce node; D, des ina ion node;
1, I-S in e e ence channel; 2, I-D in e e ence channel; h, PB-S channel; g, S-D channel.
In o ma ion
ansmission (IT)
SD
ID
EH a S
αT(1-α)T
T
Figu e 2. EH and in o ma ion p ocessing. T, he whole ansmission block.
2.1. Ene gy Ha es ing
In he i s phase, he ecei ed signal a node S du ing he EH phase is o mula ed as:
1
s
bis
yhx sn=++ (1)
whe e b
x
is he ansmi signal a he PB,
{
}
2
bB
x
PΕ=
,
{
}
Ε•is he expec a ion ope a o , PB is he
ansmi powe o he powe beacon,
s
n is he addi i e whi e Gaussian noise (AWGN) wi h ze o-
mean and a iance N0, Si is he ansmi signal a he in e e e ,
{
}
2
iI
s
PΕ=
, and PI is he ansmi
powe o he in e e e .
The o al ecei ed ene gy du ing he i s phase a node S is o mula ed as:
22
1sB B I I
EhPT PT
η
αη α
=+ (2)
whe e
{
}
0,1
BI
ηη
<<
is he ene gy con e sion e iciency o he PB and he in e e e , espec i ely.
We assume ha BI
η
ηη
==. Hence, he a e age ansmi powe a node S could be ob ained by
he ollowing equa ion:
Figu e 1.
Sys em model. PB, powe beacon-assis ed node; EH, ene gy ha es ing; I, co-channel
in e e ence om he en i onmen ; IT, in o ma ion ansmission; S, sou ce node; D, des ina ion node;
1, I-S in e e ence channel; 2, I-D in e e ence channel; h, PB-S channel; g, S-D channel.
Ene gies 2019, 12, x FOR PEER REVIEW 3 o 15
2. Sys em Model
In his wo k, we conside a PB EH in an HD communica ion ne wo k. F om Figu e 1, we deno e
ha S and D a e he sou ce node and he des ina ion node, espec i ely, PB deno es a powe beacon-
assis ed node, and I deno es he co-channel in e e ence om he en i onmen . Le 1 and 2 ep esen
he I-S and I-D in e e ence channels, espec i ely. Le h and g deno e he PB-S and S-D channels.
Assume ha all channels a e Rayleigh block ading channels. As d awn in Figu e 2, he whole
ansmission block (T) can be di ided in o pa s. Le T and 0 < α < 1 deno e he whole symbol du a ion
and he TS ac o , espec i ely. S and I sca enge ene gy om he adio equency signal ecei ed
om he PB node S and node I, espec i ely, du ing αT. Mo eo e , he emaining (1 − α)T is spen on
signal ansmission om node S o node D, and he signal om node I o node D. No e ha all he
ene gy ha es ed a node S and node I is consumed o o wa ding sou ce in o ma ion o node D.
IT
EH
PB S D
I
Figu e 1. Sys em model. PB, powe beacon-assis ed node; EH, ene gy ha es ing; I, co-channel
in e e ence om he en i onmen ; IT, in o ma ion ansmission; S, sou ce node; D, des ina ion node;
1, I-S in e e ence channel; 2, I-D in e e ence channel; h, PB-S channel; g, S-D channel.
In o ma ion
ansmission (IT)
SD
ID
EH a S
αT(1-α)T
T
Figu e 2. EH and in o ma ion p ocessing. T, he whole ansmission block.
2.1. Ene gy Ha es ing
In he i s phase, he ecei ed signal a node S du ing he EH phase is o mula ed as:
1
s
bis
yhx sn=++ (1)
whe e b
x
is he ansmi signal a he PB,
{
}
2
bB
x
PΕ=
,
{
}
Ε•is he expec a ion ope a o , PB is he
ansmi powe o he powe beacon,
s
n is he addi i e whi e Gaussian noise (AWGN) wi h ze o-
mean and a iance N0, Si is he ansmi signal a he in e e e ,
{
}
2
iI
s
PΕ=
, and PI is he ansmi
powe o he in e e e .
The o al ecei ed ene gy du ing he i s phase a node S is o mula ed as:
22
1sB B I I
EhPT PT
η
αη α
=+ (2)
whe e
{
}
0,1
BI
ηη
<<
is he ene gy con e sion e iciency o he PB and he in e e e , espec i ely.
We assume ha BI
η
ηη
==. Hence, he a e age ansmi powe a node S could be ob ained by
he ollowing equa ion:
Figu e 2. EH and in o ma ion p ocessing. T, he whole ansmission block.
2.1. Ene gy Ha es ing
In he i s phase, he ecei ed signal a node S du ing he EH phase is o mula ed as:
ys=hxb+ 1si+ns(1)
whe e
xb
is he ansmi signal a he PB,
En|xb|2o=PB
,
E{•}
is he expec a ion ope a o , P
B
is he
ansmi powe o he powe beacon,
ns
is he addi i e whi e Gaussian noise (AWGN) wi h ze o-mean
and a iance N
0
,S
i
is he ansmi signal a he in e e e ,
En|si|2o=PI
, and P
I
is he ansmi powe o
he in e e e .
The o al ecei ed ene gy du ing he i s phase a node S is o mula ed as:
Es=ηB|h|2PBαT+ηI 1
2PIαT(2)
whe e 0 <ηB,ηI<1 is he ene gy con e sion e iciency o he PB and he in e e e , espec i ely.
Ene gies 2019,12, 2579 4 o 14
We assume ha
ηB=ηI=η
. Hence, he a e age ansmi powe a node S could be ob ained
by he ollowing equa ion:
Ps=Es
(1−α)T=η|h|2PBαT+η 1
2PIαT
(1−α)T=κPB|h|2+ 1
2PI(3)
whe e κ=ηα
1−α.
2.2. In o ma ion T ansmission
In he second phase, node S ansmi s he signal o node D, and he ecei ed signal, y
D
, a he
des ina ion is o mula ed as:
yD=gxs+ 2si+nd=gxs
|{z}
signal
+ 2si+nd
| {z }
noise
(4)
whe e we ha e En|xs|2o=Ps, and ndis he AWGN wi h ze o-mean and a iance N0.
F om Equa ion (4), he end o end signal o in e e ence plus noise a io (SINR) was calcula ed as
he ollowing:
γe2e=
Esignal
2
En|noise|2o=g
2Ps
 2
2PI+N0
. (5)
Subs i u ing Equa ion (3) in o Equa ion (5) and assuming ha he powe o in e e e noise is e y
la ge, so PB≈PI, we ha e:
γe2e=
κg
2PB|h|2+ 1
2PI
 2
2PI+N0
=
κg
2∆|h|2+ 1
2
 2
2∆+1
(6)
whe e ∆=PB
N0=PI
N0.
3. The Sys em Pe o mance
3.1. Ou age P obabili y
Based on he sys em model in he abo e sec ion, we de i ed he OP h oughpu pe o mance and
EE o he p oposed sys em.
F om Equa ion (6) we ob ain he OP o he model sys em as he ollowing:
OP =P (γe2e< γ0)=P "κ|g|2∆|h|2+| 1|2
| 2|2∆+1< γ0#
=P (XY < γ0)=
∞
R0
FYγ0
XX=x X(x)dx
(7)
whe e
γ0=
2
2R−
1 is he h eshold o he sys em, R is he sou ce a e,
X=|h|2+ 1
2=ϕ1+
ϕ2,Y=κ|g|2∆
| 2|2∆+1=κϕ3∆
1+∆ϕ4, and ϕ1=|h|2,ϕ2= 1
2,ϕ3=g
2,ϕ4= 2
2.
In o de o calcula e he p obabili y in Equa ion (7), we ha e o de e mine he p obabili y densi y
unc ion (PDF) and he cumula i e densi y unc ion (CDF) o Xand Y, espec i ely, as in Lemma 1 and
Lemma 2.
Ene gies 2019,12, 2579 5 o 14
Lemma 1. The CDF o Y can be compu ed as:
FY(a) = P (Y<a)=P κϕ3∆
1+∆ϕ4<a=P ϕ3<a
κ∆+aϕ4
κ
=
∞
R0
ϕ4(ϕ4)dϕ4
a
κ∆+aϕ4
κ
R0
ϕ3(ϕ3)dϕ3=
∞
R0
Fϕ3a
κ∆+aϕ4
κ ϕ4(ϕ4)dϕ4
=
∞
R0
ϕ4(ϕ4)dϕ4−1
λ4
∞
R0
exp−a
κ∆λ3−aϕ4
κλ3exp−ϕ4
λ4dϕ4
=1−κλ3exp−a
κ∆λ3
aλ4+κλ3
(8)
whe e λ3,λ4a e he mean o he andom a iables (RVs) ϕ3,ϕ4, espec i ely.
Lemma 2. The CDF o X can be exp essed as:
FX(a) = P [(ϕ1+ϕ2)<a]=P [ϕ1<a−ϕ2]
=
a
R0
ϕ2(ϕ2)dϕ2
a−ϕ2
R0
ϕ1(ϕ1)dϕ1=
a
R0
Fϕ1(a−ϕ2) ϕ2(ϕ2)dϕ2
=1
λ2
a
R0h1−exp−a−ϕ2
λ1iexp−ϕ2
λ2dϕ2
=1−exp−a
λ2−exp−a
λ1
λ2
a
R0
expϕ2h1
λ1−1
λ2idϕ2
(9)
whe e λ1,λ2a e he mean o he RVs ϕ1,ϕ2, espec i ely.
In his si ua ion, we in es iga ed wo cases as ollows:
Case 1: We assume ha λ1=λ2=λ.
F om Equa ion (9), we ob ain:
FX(a) = 1−exp−a
λ−aexp−a
λ
λ. (10)
The e o e, he PDF can be ob ained as:
X(a) = ∂FX(a)
∂a=aexp−a
λ
λ2(11)
Case 2: We assume ha λ1,λ2.
Simila ly, we can ob ain he CDF and he PDF o Xas ollows:
FX(a) = 1−exp−a
λ2−λ1
λ2−λ1"exp−a
λ2−exp −a
λ1!#. (12)
X(a) = 1
λ2−λ1"exp−a
λ2−exp −a
λ1!#. (13)
3.1.1. Exac Analysis
Case 1: λ1=λ2=λ.

Ene gies 2019,12, 2579 6 o 14
Combining Equa ion (8), Equa ion (11) and applying hem o Equa ion (7) allows he OP o be
compu ed as:
OP1=
∞
R0
FYγ0
XX=x X(x)dx =1−
∞
R0
κλ3exp−γ0
xκ∆λ3
γ0
xλ4+κλ3
×xexp(−x
λ)
λ2dx
=1−κλ3
λ2
∞
R0
x2
κλ3x+γ0λ4×exp−γ0
κ∆λ3x×exp−x
λdx
. (14)
Case 2: λ1,λ2.
Combining Equa ion (8) and Equa ion (13), we can ob ain he OP, in his case as ollows:
OP2=1−
∞
R0
κλ3exp−γ0
xκ∆λ3
γ0
xλ4+κλ3
×n1
λ2−λ1hexp−x
λ2−exp−x
λ1iodx
=1−κλ3
λ2−λ1
∞
R0
xexp−γ0
κ∆λ3x
κλ3x+γ0λ4×nexp−x
λ2−exp−x
λ1odx
. (15)
3.1.2. Asymp o ic Analysis
The high SINR egime and he end o end SINR om Equa ion (6) can be app oxima ed as:
γ∞
e2e≈
κg
2|h|2+ 1
2
 2
2=ZX (16)
whe e X=|h|2+ 1
2=ϕ1+ϕ2,Z=κ|g|2
| 2|2=κϕ3
ϕ4.
Lemma 3. The CDF o Z can be calcula ed using he equa ion:
FZ(a) = P (Z<a)=P κϕ3
ϕ4<a=P ϕ3<aϕ4
κ
=
∞
R0
ϕ4(ϕ4)dϕ4
aϕ4
κ
R0
ϕ3(ϕ3)dϕ3
=1−1
λ4
∞
R0
exp−aϕ4
κλ3×exp−ϕ4
λ4dϕ4
=1−κλ3
aλ4+κλ3
. (17)
Case 1: λ1=λ2=λ.
F om Equa ions (11), (16), and (17), he OP can be compu ed as:
OP∞
1=P γ∞
e2e< γ0=P "κ|g|2|h|2+| 1|2
| 2|2< γ0#
=P (XZ < γ0)=
∞
R0
FZγ0
XX=x X(x)dx
=1−1
λ2
∞
R0
x2
x+γ0λ4
κλ3
×exp−x
λdx
. (18)
Ene gies 2019,12, 2579 7 o 14
Applying Equa ion (3.353,5) om he able o in eg als [29], Equa ion (18) can be ew i en as:
OP∞
1=γ0λ4λ
κλ3
+ γ0λ4
κλ3λ!2
exp γ0λ4
κλ3λ!Ei −γ0λ4
κλ3λ!(19)
whe e Ei(−z)=−
∞
Rz
e− −1d is he exponen ial in eg al unc ion.
Case 2:λ1,λ2.
In his case, based on Equa ions (13), (16), and (17), he OP can be calcula ed as:
OP∞
2=1−1
λ2−λ1
∞
R0
x
x+γ0λ4
κλ3
×nexp−x
λ2−exp−x
λ1odx
=1−1
λ2−λ1
∞
R0
xexp−x
λ2
x+γ0λ4
κλ3
dx +1
λ2−λ1
∞
R0
xexp−x
λ1
x+γ0λ4
κλ3
dx.
(20)
Simila o he p e ious case, we apply Equa ion (3.353,5) om he able o in eg als [
29
], hus
Equa ion (19) can be o mula ed as ollows:
OP∞
2=1−γ0λ4
κλ3(λ2−λ1)expγ0λ4
κλ3λ2Ei−γ0λ4
κλ3λ2−λ2
λ2−λ1
+γ0λ4
κλ3(λ2−λ1)expγ0λ4
κλ3λ1Ei−γ0λ4
κλ3λ1+λ1
λ2−λ1
=γ0λ4
κλ3(λ2−λ1)nexpγ0λ4
κλ3λ1Ei−γ0λ4
κλ3λ1−expγ0λ4
κλ3λ2Ei−γ0λ4
κλ3λ2o
. (21)
3.2. Ene gy E iciency Analysis
The EE is de ined as a a io o o al in o ma ion a e Cand o al powe consump ion ETin [18]:
EE =C
ET
(22)
whe e
C=(1−α)T
2log2(
1
+γe2e)
and
ET=[2αT+ (1−α)T]P
ε+PCT
, in which
P=PB=PI
,
and
ε
and
PC
deno e he powe ampli ie e iciency o he powe beacon and ci cui powe
consump ion, espec i ely.
Finally, we ob ain he EE as shown in he equa ion below:
EE =
(1−α)
2log2(1+γe2e)
(T+αT)P/ε+PCT=
(1−α)
2log2(1+γe2e)
(1+α)P/ε+PC
. (23)
In o de o analyze he EE, we conside ed calcula ing he a e age o al in o ma ion a e in wo
cases: λ1=λ2=λand λ1,λ2.
The a e age o al in o ma ion a e can be exp essed as [30]:
Ca g =(1−α)
2 ln 2
∞
Z
0
1−Fγe2e(γ0)
1+γ0
dγ0. (24)
Case 1: λ1=λ2=λ.
a. Exac Analysis
Ene gies 2019,12, 2579 8 o 14
Subs i u ing Equa ion (14) in o Equa ion (24), we ob ain Ca g as ollows:
C1_a g =(1−α)
2 ln 2
∞
R0
1−Fγe2e(γ0)
1+γ0dγ0
=(1−α)κλ3
2λ2ln 2
∞
R0
∞
R0
x2
(κλ3x+γ0λ4)(1+γ0)×exp−γ0
κ∆λ3x×exp−x
λdxdγ0
. (25)
Nex , subs i u ing Equa ion (25) in o Equa ion (22), inally, he a e age EE can be gi en as:
EE1_a g =
(1−α)κλ3
2λ2ln 2
∞
R0
∞
R0
x2
(κλ3x+γ0λ4)(1+γ0)×exp−γ0
κ∆λ3x×exp−x
λdxdγ0
(1+α)P/ε+PC
. (26)
b. Asymp o ic Analysis
Subs i u ing Equa ion (19) in o Equa ion (24), he a e age o al in o ma ion a e can be exp essed
as:
C∞
1_a g =(1−α)
2 ln 2
∞
Z
0
1−






γ0λ4λ
κλ3
+ γ0λ4
κλ3λ!2
exp γ0λ4
κλ3λ!Ei −γ0λ4
κλ3λ!






dγ0
1+γ0
. (27)
Combining Equa ion (25) wi h Equa ion (27), he EE in Case 1 can be ob ained as
EE∞
1=
(1−α)
2 ln 2
∞
R01−γ0λ4λ
κλ3+γ0λ4
κλ3λ2expγ0λ4
κλ3λEi−γ0λ4
κλ3λ dγ0
1+γ0
(1+µ)P/ε+PC
. (28)
Case 2: λ1,λ2.
Simila o Case 1, we can calcula e he EE o he exac and he asymp o ic analyses, espec i ely,
as ollows:
EE2_a g =
(1−α)κλ3
2(λ2−λ1)ln 2
∞
R0
∞
R0
xexp−γ0
κ∆λ3x
(κλ3x+γ0λ4)(1+γ0)×nexp−x
λ2−exp−x
λ1odxdγ0
(1+α)P/ε+PC
(29)
EE∞
2_a g =
(1−α)
2 ln 2
∞
R01−γ0λ4
κλ3(λ2−λ1)nexpγ0λ4
κλ3λ1Ei−γ0λ4
κλ3λ1−expγ0λ4
κλ3λ2Ei−γ0λ4
κλ3λ2o dγ0
1+γ0
(1+α)P/ε+PC
(30)
4. Resul s and Discussion
In his sec ion, we used he Mon e Ca lo simula ion o e i y he accu acy o he analysis
exp essions om he p e ious sec ion. Fo each simula ion, we i s p o ided he g aphs o he OP
and he EE ob ained by he analy ical o mulas. Secondly, we gene a ed plo s o he same OP and EE
cu es ha esul ed om he Mon e Ca lo simula ion. To do his, we gene a ed 10
5
andom samples o
each channel gain, which we e Rayleigh dis ibu ed. The analy ical cu e and he simula ion should
ma ch oge he o e i y he accu acy o ou analysis [30–33].
In Figu e 3, we plo ed he e ec o he ime swi ching ac o
α
on he OP in wo cases—Case 1 and
Case 2, espec i ely. In hese cases, we se he p ima y sys em pa ame e s as R ={1,3} bps/Hz,
∆
=5 dB,
and
η
=0.8. F om he esea ch esul s, he OP o he p oposed sys em dec eased wi h inc easing
α
om 0 o 1, and he OP o R =1 bps/Hz was no be e han ha wi h R =3 bps/Hz. I can be seen
ha he OP was be e wi h he highe R, and ha he OP o Case 2 was be e han ha o Case 1.
Mo eo e , he OP e sus
∆
is illus a ed in Figu e 4wi h he exac and asymp o ic exp essions. In his
Ene gies 2019,12, 2579 9 o 14
case, we se R =1 bps/Hz and
α
=0.5. Simila o Figu e 3, we conside ed wo cases, Case 1 and Case 2,
in Figu e 4. We obse ed ha he OP dec eased signi ican ly wi h inc easing
∆
om
−
5 o 15 dB and
hen con e ged in o he asymp o ic OP wi h
∆
om 15 o 25, as shown in Figu e 4. Once again, we
saw ha he OP o Case 2 was be e han ha o Case 1. F om he esul s in Figu es 3and 4, we can
conclude ha all he simula ion and analy ical esul s a e he same o all alues o αand ∆.
Ene gies 2019, 12, x FOR PEER REVIEW 9 o 15
()
04 04 04 04 04 0
32 1 31 31 32 32 0
0
2_
(1 ) 1expEiexpEi
2ln2 1
(1 ) /
a g
C
d
EE PP
γ
λ
γ
λ
γ
λ
γ
λ
γ
λ
γ
α
κλ λ λ κλ λ κλ λ κλ λ κλ λ
γ
αε
∞
∞



  
− 
−−−−

 
  
−+


  




=++

(30)
4. Resul s and Discussion
In his sec ion, we used he Mon e Ca lo simula ion o e i y he accu acy o he analysis
exp essions om he p e ious sec ion. Fo each simula ion, we i s p o ided he g aphs o he OP
and he EE ob ained by he analy ical o mulas. Secondly, we gene a ed plo s o he same OP and EE
cu es ha esul ed om he Mon e Ca lo simula ion. To do his, we gene a ed 105 andom samples
o each channel gain, which we e Rayleigh dis ibu ed. The analy ical cu e and he simula ion
should ma ch oge he o e i y he accu acy o ou analysis [30–33].
In Figu e 3, we plo ed he e ec o he ime swi ching ac o α on he OP in wo cases—Case 1
and Case 2, espec i ely. In hese cases, we se he p ima y sys em pa ame e s as R = {1,3} bps/Hz, Δ
= 5 dB, and η = 0.8. F om he esea ch esul s, he OP o he p oposed sys em dec eased wi h
inc easing α om 0 o 1, and he OP o R = 1 bps/Hz was no be e han ha wi h R = 3 bps/Hz. I can
be seen ha he OP was be e wi h he highe R, and ha he OP o Case 2 was be e han ha o
Case 1. Mo eo e , he OP e sus Δ is illus a ed in Figu e 4 wi h he exac and asymp o ic exp essions.
In his case, we se R = 1 bps/Hz and α = 0.5. Simila o Figu e 3, we conside ed wo cases, Case 1 and
Case 2, in Figu e 4. We obse ed ha he OP dec eased signi ican ly wi h inc easing Δ om −5 o 15
dB and hen con e ged in o he asymp o ic OP wi h Δ om 15 o 25, as shown in Figu e 4. Once
again, we saw ha he OP o Case 2 was be e han ha o Case 1. F om he esul s in Figu es 3 and
4, we can conclude ha all he simula ion and analy ical esul s a e he same o all alues o α and
Δ.
Figu e 3. OP e sus α. OP, ou age p obabili y.
Figu e 3. OP e sus α. OP, ou age p obabili y.
Ene gies 2019, 12, x FOR PEER REVIEW 10 o 15
Figu e 4. OP e sus Δ.
Fu he mo e, he OP o he model sys em e sus he ene gy con e sion coe icien η o he wo
cases, Case 1 and Case 2, is shown in Figu e 5 wi h Δ = {1,5} dB, R = 1 bps/Hz and α = 0.5. As he
esul s a e he same in he abo e igu es, we de e mined ha he OP o he p oposed sys em
dec eased c ucially while η a ied om 0 o 1 and ha he OP o Case 2 was be e han ha o Case
1. Also, all simula ion esul s ag eed well wi h he analy ical esul s wi h a ying η.
Figu e 5. OP e sus η.
We in es iga ed he in luence o he λ1 = λ2 = λ on Case 1 o he OP o he model sys em, as
illus a ed in Figu e 6. He e, we se Δ = {1,5,10} dB, η = 0.8, α = 0.5, and R = 1 bps/Hz. F om he esul s,
we saw ha he OP ell wi h he ising λ, and he OP was be e wi h a highe alue o Δ. Mo eo e ,
he e ec o λ1 ≠ λ2 on he OP o he model sys em in connec ion wi h λ1 and λ2 is p esen ed in Figu e
7 wi h he p ima y sys em pa ame e s as Δ = {1,5,10} dB. In his si ua ion, bo h λ1 and λ2 a ied om
0 o 4. F om he esul s, we can conclude ha he OP o he sys em alls when λ1 and λ2 inc ease. The
Figu e 4. OP e sus ∆.
Fu he mo e, he OP o he model sys em e sus he ene gy con e sion coe icien
η
o he wo
cases, Case 1 and Case 2, is shown in Figu e 5wi h
∆
={1,5} dB, R =1 bps/Hz and
α
=0.5. As he
esul s a e he same in he abo e igu es, we de e mined ha he OP o he p oposed sys em dec eased
c ucially while η a ied om 0 o 1 and ha he OP o Case 2 was be e han ha o Case 1. Also, all
simula ion esul s ag eed well wi h he analy ical esul s wi h a ying η.